Problem: Simplify; express your answer in exponential form. Assume $p\neq 0, k\neq 0$. $\dfrac{{p^{-2}k^{-1}}}{{(p^{4}k^{-2})^{-1}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${p^{-2}k^{-1} = p^{-2}k^{-1}}$ On the left, we have ${p^{-2}}$ to the exponent ${1}$ . Now ${-2 \times 1 = -2}$ , so ${p^{-2} = p^{-2}}$ Apply the ideas above to simplify the equation. $\dfrac{{p^{-2}k^{-1}}}{{(p^{4}k^{-2})^{-1}}} = \dfrac{{p^{-2}k^{-1}}}{{p^{-4}k^{2}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{-2}k^{-1}}}{{p^{-4}k^{2}}} = \dfrac{{p^{-2}}}{{p^{-4}}} \cdot \dfrac{{k^{-1}}}{{k^{2}}} = p^{{-2} - {(-4)}} \cdot k^{{-1} - {2}} = p^{2}k^{-3}$